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80D. Liu, P. Ning and W. Du•If Pos(i)<Pos(j) and (Pos(i) + Pos(j)) mod 2 = 0, nodeihas thepre-distributed keyH(Kj||i),while nodejcan compute this keyeasily.•If Pos(i)>Pos(j) and (Pos(i) + Pos(j)) mod 2 = 1, nodeihas thepre-distributed keyH(Kj||i),while nodejcan compute this keyeasily.•If Pos(i)>Pos(j) and (Pos(i)+Pos(j)) mod 2 = 0, nodeican computethe shared keyH(Ki||j), while nodejhas already been pre-distributedwith this key.From the above discussion, we can clearly see that a sensor node canimmediately determine the direct key shared with another sensor node inthe same group.There is no additional communication overhead.Sinceany two sensor nodes in the same group shared a unique key, there is noneed to implement the path key establishment inside a group.In addition, we can also see that every shared key between two sensornodes is only known by these two related sensor nodes. Hence, the com-promise of sensor nodes does not lead to the compromise of any direct keybetween two non-compromised sensor nodes. As a result, the above schemeguarantees theperfect security propertyin the presence of node compromiseattacks.Now let us estimate the storage space for the keying materials at sensornodes.Consider any sensor nodeiin a deployment groupG.There arePos(i)−1 sensor nodes with smaller IDs andn−Pos(i) sensor nodes withlarger IDs. When Pos(i) is an odd number, we can easily see that nodeiwill get pre-distributedPos(i)−12keys for the sensor nodes with smaller IDs,andn−P os(i)2keys for the sensor nodes with larger IDs.Therefore, theoverall number of pre-distributed pairwise keys to nodeifor the deploymentgroupGisPos(i)−12+n−Pos(i)2≈n2.(2)Similarly, if Pos(i) is even, the number of pre-distributed pairwise keysto nodeiis approximatelyn2. When nodeibelongs to a cross groupG,we can also estimate the number of pre-distributed pairwise keys asm2in a similar way.Thus, the total number of pairwise keys that are pre-distributed to a sensor node for both of its deployment group and cross

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